After spending more than two weeks reading well over 750 pages while checking the following documents…
- "Sponge Functions"
- "Cryptographic sponge functions"
- "Security Analysis of Extended Sponge Functions"
- "Cryptographic Hash Functions: Recent Design Trends and
- "On the Implementation Aspects of Sponge-based
Authenticated Encryption for Pervasive Devices"
- "On the security of the keyed sponge construction"
- "Keccak and the SHA-3 Standardization"
- "Computer Science and Its Applications: Csa 2012"
…it turns out that the probability of the claim can only be "proven" theoretically. In other words: currently, there is no way to actually and/or practically verify the probability of the claim.
The current status quo states that collisions are theoretically proven to be very unlikely in a cryptographic sense. I guess that — as with all cryptographic theories — only time can tell if that actually provides a problem or not.
But getting back to my question: no, it can't actually be verified currently.
I don't expect this to change anytime soon either, as the theoretical proof that collisions are very unlikely in a cryptographic sense, seems to be strong enough to regard sponge functions as (let's just call it) "secure". In fact, if you check the linked documents, you will find ample information on how to replicate the theoretical proofs in relation to collision probabilities using sponge functions under a multitude of conditions. As far as I checked, cross-checked, and even practically used those formulas to verify things myself, I got convinced that collisions are indeed very (very) unlikely.
I hope you don't mind that I'm sparing you a copy-and-paste job of more than a dozen of pages full of math code. Those who want to dive in deeper, can fetch the multitude of related formulas and a truckload of additional, interesting information from the linked documents I listed above.